Editing Heaviside step function (section)

== Fourier transform ==
The [[Fourier transform]] of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have
<math display="block">\hat{H}(s) = \lim_{N\to\infty}\int^N_{-N} e^{-2\pi i x s} H(x)\,dx = \frac{1}{2} \left( \delta(s) - \frac{i}{\pi} \operatorname{p.v.}\frac{1}{s} \right).</math>

Here {{math|p.v.{{sfrac|1|''s''}}}} is the [[distribution (mathematics)|distribution]] that takes a test function {{mvar|φ}} to the [[Cauchy principal value]] of <math>\textstyle\int_{-\infty}^\infty \frac{\varphi(s)}{s} \, ds</math>. The limit appearing in the integral is also taken in the sense of (tempered) distributions.